A chord changes a sector into a segment
A sector is bounded by two radii and an arc. A segment is bounded by a chord and its arc; the triangle between the radii must therefore be removed from the sector.
Learning goals
- Distinguish sectors and segments, including minor and major regions.
- Determine the area of the triangle formed by two radii and a chord.
- Calculate minor and major segment areas.
Concept lesson
Sector area
Sector area is theta/360 multiplied by pi r squared. Keep the exact central angle attached to the required arc.
Triangle area toolkit
Use one-half base times perpendicular height, an equilateral-triangle formula, or split an isosceles triangle into right triangles according to the supplied angle.
Read the boundary first
Two radii plus an arc enclose a sector; one chord plus an arc enclose a segment. The smaller regions are minor, while their whole-circle complements are major.
Segment subtraction and complements
Minor segment area equals minor sector area minus triangle area. Major segment area equals circle area minus minor segment area; major sector area equals circle area minus minor sector area.
Triangle inside a 60-degree sector
When OA=OB=r and angle AOB is 60 degrees, triangle OAB is equilateral. Calculate its area, calculate the 60-degree sector, then subtract to obtain the minor segment.
A 120-degree segment
Bisect the isosceles triangle with a perpendicular from O to chord AB. Use the resulting 60-degree right triangle for height and half-chord, then subtract triangle area from the 120-degree sector.
Sector or segment classifier
Trace each shaded boundary and classify it before calculating; explicitly name whether the chord or two radii enclose the region.
Write the region equation before numbers
Write minor segment=sector-triangle or major segment=circle-minor segment before substituting any values.
Choose a triangle-area route
From the central angle, decide whether the radius triangle is right, equilateral or must be bisected; calculate the perpendicular height without inventing a chord length.
Minor and major region chain
Calculate sector and triangle components, obtain a minor segment, then use the circle complement for a major region; verify that the two segment areas sum to the circle area.
Three-line segment solution skeleton
Show sector area, triangle area and their correct combination as separately labelled steps; retain the specified pi and radical approximation policy.
Segment-area mastery
Classify a shaded region, find the internal triangle area, calculate minor and major segment measures, and verify by a whole-circle complement.